Align plane to ground
Create an empty and turn on face-snapping with influence on move and rotate before snapping it to your rotated plane(empty is aligned):
  

parent the rotated plane to the empty and finally set the rotation of the empty as 0, 0, 0 before clearing the parent-relation:


explanation: the problem with the snapping without the empty is that there are no object rotations in Edit Mode, but only single vertices and their locations. If your rotation in Object Mode is applied (0,0,0) or not represented in the Edit Mode transformation of your rotated plane, then there is no difference to align to an other object also having 0,0,0 rotation in Object Mode. Only the transformation of the of the new empty (0, 0, 0) to the normal of your plane allows to rotate the plane in relation to this newly formed difference.